The fields \(\Phi \) , \( V\)  and \( X\)  satisfy the static no-particle WS equations, which are the Euler-Lagrange equations for the static WS energy functional originating in (REF )For a discussion of the the time-dependent theory and a derivation of the energy functional (REF ) see [1]}, [2]}, [3]}, [4]} and Appendix .
\( E_{N}^{WS} (\Phi , Q) := \int _{N}\big (\Vert  \nabla _{Q}\Phi \Vert _{\Omega ^1_{2}}^{2} + \frac{1}{2} \lambda (\Vert \Phi \Vert _{2}^2-\varphi _0^2)^2 + \frac{1}{2}\Vert  F_Q\Vert _{\Omega ^2_{\mathfrak {u}(2)}}^2\big ), \) 
